where \(I\)  is an interval of \(\partial \mathbb {D}\)  and \(|I|\)  denotes the length of \(I\) . If \(\mu \)  satisfies \(\lim _{|I|\rightarrow 0}\frac{\mu (S(I))}{|I|^s}=0\) , we say that \(\mu \)  is a vanishing s-Carleson measure. It is well known [1]} that for \(0<p\le q<\infty \) , \(\mu \)  is a \(\frac{2q}{p}\) -Carleson measure if and only if
there exists a positive constant \(C\)  such that the inequality
\( \left\lbrace \int _{\mathbb {D}}|f(z)|^qd\mu (z)\right\rbrace ^{1/q}\le C\Vert f\Vert _{A^p}\) 
